Ordering-Aware Theory of Trotter Error
Abstract
Product formulas are a basic tool for simulating quantum many-body dynamics, but a fundamental degree of freedom in their application---the ordering of Hamiltonian terms---has remained largely unexplored. Here we systematically study such ordering effects, quantifying their impact on simulation accuracy and exploiting them to design more accurate product-formula simulations. We develop an efficiently computable, ordering-sensitive bound on the finite-step-size Trotter error for geometrically local Hamiltonians for arbitrarily high-order product formulas. For translation-invariant one-dimensional chains, our bounds are close to exact second-, fourth-, and sixth-order Trotter errors in small-system tests, while on large-scale systems they improve on previous higher-order analytical bounds by orders of magnitude. We further use the local structure of the error bounds and estimates to optimize ordering efficiently, reducing the long-chain problem to a minimum-mean-weight-cycle problem on a weighted de Bruijn graph. For random translation-invariant chains, sequential (staircase-like) orderings consistently outperform standard brickwall circuits; when circuit depth is restricted, the optimizer instead favors wave-like parallel-sequential orderings, which preserve much of the accuracy advantage over brickwall circuits. These results establish ordering as a useful degree of freedom for both understanding and improving product-formula simulation.
Related
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BibTeX
@misc{liu2026orderingaware,
title = {{Ordering-Aware Theory of Trotter Error}},
author = {Zhenning Liu and Zhi-Yuan Wei and Michael Foss-Feig and Alexey V. Gorshkov and Andrew M. Childs},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03672},
eprint = {2610.03672},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03672},
}
RIS
TY - UNPB
TI - Ordering-Aware Theory of Trotter Error
AU - Zhenning Liu
AU - Zhi-Yuan Wei
AU - Michael Foss-Feig
AU - Alexey V. Gorshkov
AU - Andrew M. Childs
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03672
UR - https://arxiv.org/abs/2610.03672
AB - Product formulas are a basic tool for simulating quantum many-body dynamics, but a fundamental degree of freedom in their application---the ordering of Hamiltonian terms---has remained largely unexplored. Here we systematically study such ordering effects, quantifying their impact on simulation accuracy and exploiting them to design more accurate product-formula simulations. We develop an efficiently computable, ordering-sensitive bound on the finite-step-size Trotter error for geometrically local Hamiltonians for arbitrarily high-order product formulas. For translation-invariant one-dimensional chains, our bounds are close to exact second-, fourth-, and sixth-order Trotter errors in small-system tests, while on large-scale systems they improve on previous higher-order analytical bounds by orders of magnitude. We further use the local structure of the error bounds and estimates to optimize ordering efficiently, reducing the long-chain problem to a minimum-mean-weight-cycle problem on a weighted de Bruijn graph. For random translation-invariant chains, sequential (staircase-like) orderings consistently outperform standard brickwall circuits; when circuit depth is restricted, the optimizer instead favors wave-like parallel-sequential orderings, which preserve much of the accuracy advantage over brickwall circuits. These results establish ordering as a useful degree of freedom for both understanding and improving product-formula simulation.
ER -